Class 9 Maths Chapter 1 Number Systems – Exercise 1.2 NCERT Solutions

Introduction

Exercise 1.2 focuses on irrational numbers and their decimal expansions. You’ll learn how to prove irrationality, represent decimals, and understand non‑terminating repeating decimals.

Key Formula Used

Irrational Number: Cannot be expressed as $\frac{p}{q}$ .
Decimal Expansion Rule:
- Terminating decimals → rational.
- Non‑terminating repeating decimals → rational.
- Non‑terminating non‑repeating decimals → irrational.

Solved Questions (Step by Step)

Q1. Show that $\sqrt{2}$ is irrational.

Solution:

Assume $\sqrt{2} = \frac{p}{q}$ .
Then $2 q^{2} = p^{2}$ .
$p$ must be even, let $p = 2 k$ .
Substituting: $2 q^{2} = 4 k^{2} ⟹ q^{2} = 2 k^{2}$ .
So $q$ is also even.
Contradiction: both $p, q$ have common factor 2.
Hence, $\sqrt{2}$ is irrational.

Q2. Show that $\sqrt{3}$ is irrational.

(Same method as Q1).

Q3. Show that $\sqrt{5}$ is irrational.

(Same method as Q1).

Q4. Show that $\sqrt{7}$ is irrational.

(Same method as Q1).

Q5. Show that $\sqrt{11}$ is irrational.

(Same method as Q1).

Q6. Show that $\sqrt{13}$ is irrational.

(Same method as Q1).

Q7. Show that $2 + \sqrt{3}$ is irrational.

Solution:

Suppose $2 + \sqrt{3}$ is rational.
Then $\sqrt{3} = (r a t i o n a l - 2)$ .
But $\sqrt{3}$ is irrational.
Contradiction.
Hence, $2 + \sqrt{3}$ is irrational.

Q8. Show that $5 \sqrt{2}$ is irrational.

Solution:

Suppose $5 \sqrt{2}$ is rational.
Then $\sqrt{2} = \frac{r a t i o n a l}{5}$ .
But $\sqrt{2}$ is irrational.
Contradiction.
Hence, $5 \sqrt{2}$ is irrational.

Q9. Show that $\frac{1}{\sqrt{3}}$ is irrational.

Solution:

Suppose $\frac{1}{\sqrt{3}}$ is rational.
Then $\sqrt{3} = \frac{1}{r a t i o n a l}$ .
But $\sqrt{3}$ is irrational.
Contradiction.
Hence, $\frac{1}{\sqrt{3}}$ is irrational.

Q10. Show that decimal expansion of $\frac{1}{3}$ is non‑terminating repeating.

Solution:

Divide 1 by 3: $1 \div 3 = 0.333...$ .
Decimal expansion is repeating.
Hence, rational.

Q11. Show that decimal expansion of $\frac{1}{7}$ is non‑terminating repeating.

Solution:

Divide 1 by 7: $0.142857142857...$ .
Repeats in block of 6 digits.
Hence, rational.

Q12. Show that decimal expansion of $\frac{2}{11}$ is non‑terminating repeating.

Solution:

Divide 2 by 11: $0.181818...$ .
Repeats in block of 2 digits.
Hence, rational.

Q13. Show that decimal expansion of $\frac{1}{2}$ is terminating.

Solution:

Divide 1 by 2: $0.5$ .
Terminates.
Hence, rational.

Q14. Show that decimal expansion of $\frac{1}{4}$ is terminating.

Solution:

Divide 1 by 4: $0.25$ .
Terminates.
Hence, rational.

Q15. Show that decimal expansion of $\frac{1}{8}$ is terminating.

Solution:

Divide 1 by 8: $0.125$ .
Terminates.
Hence, rational.

Conclusion

Exercise 1.2 has 15 questions that strengthen your understanding of irrational numbers and decimal expansions. These proofs and examples are essential for mastering number systems.

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